Powerful Fibonacci polynomials over finite fields
Number Theory
2026-01-07 v1
Abstract
Bugeaud, Mignotte, and Siksek proved that the only perfect powers in Fibonacci sequence are 0, 1, 8, and 144. In this paper, we study the polynomial analogue of the problem. Especially, we give a complete characterization of the Fibonacci polynomials that are perfect powers or powerful over finite fields, where there are infinitely many of them. We also give similar characterizations for some of Horadam's generalized Lucas polynomial sequences, which include Fibonacci, Lucas, Chebyshev, and Jacobsthal polynomials.
Cite
@article{arxiv.2601.02664,
title = {Powerful Fibonacci polynomials over finite fields},
author = {Graeme Bates and Ryan Jesubalan and Seewoo Lee and Jane Lu and Hyewon Shim},
journal= {arXiv preprint arXiv:2601.02664},
year = {2026}
}
Comments
14 pages, 5 tables